# Analysis Seminar

#### Remarks on Onsager's Conjecture and Anomalous Dissipation on domains with and without boundaries

**Speaker:**
Theodore Drivas, Princeton University

**Location:**
Warren Weaver Hall 1302

**Date:**
Thursday, April 5, 2018, 11 a.m.

**Synopsis:**

We first discuss the inviscid limit of the global energy dissipation of

Leray solutions of incompressible Navier-Stokes on the torus. Assuming

that the solutions have Besov norms bounded uniformly in viscosity, we

establish an upper bound on energy dissipation. As a consequence, Onsager-type

"quasi-singularities" are required in the Leray solutions, even if the total

energy dissipation is o(ν) in the limit ν → 0. Next, we discuss an extension

of Onsager's conjecture for domains with solid boundaries. We give a localized

regularity condition for energy conservation of weak solutions of the Euler

equations assuming Besov regularity of the velocity with σ>1/3 for any U⋐Ω

and, on an arbitrary thin layer around the boundary, boundedness of velocity,

pressure and continuity of the wall-normal velocity. We also prove that the global

viscous dissipation vanishes in the inviscid limit for Leray-Hopf solutions of the

Navier-Stokes equations under the similar assumptions, but holding uniformly in a

vanishingly thin viscous boundary layer. Finally, if a strong Euler solution exists,

we show that equicontinuity at the boundary within a O(ν) strip alone suffices to

conclude the absence of anomalous dissipation.

The first part of the talk concerns joint work with G. Eyink, the second with H.Q. Nguyen.